1

# Introduction

The mortality of a modern developed society is largely the mortality of old age, and most of the information about it may be summarised by two parameters, the first indicating the age near which most of the deaths occur, and the second indicating the spread of deaths about that age.

The broad shape of most, if not all, modern life tables is still that of Gompertz (1825), even if precise fits are not obtainable. Under the Gompertz “law”, the force of mortality (or hazard function) at age x may be written in the form

_{x }= B e^{kx }.

(1)

The constant B indicates the base level of mortality in the life table, whilst k shows the increase in mortality with age. Since the force of mortality at the mode of the curve of deaths coincides with the ageing parameter k, formula (1) may be re-written as

_{x }= k exp{(x-m)k},

where m is the mode of the curve of deaths.

(2)

The Gompertz distribution with hazard function (1) over the entire range (-,) has standard deviation (6)/k (Pollard and Valkovics, 1992), and since very little of the density is located on the negative part of the axis for values of B and k encountered with human life tables, the standard deviation of the time to death under the Gompertz “law” of a life aged zero is close to (6)/k. The mode m and ageing parameter k of the Gompertz “law” (2) are therefore the two essential parameters summarising modern mortality, providing measures of location and spread respectively.

How well do the parameters m and k summarise the overall pattern of mortality in a modern developed population?

To find an answer to this question, the mode and force of mortality at the mode were obtained for seven modern life tables: the Australian Life Tables (Males) 1990-92, the Australian Life Table (Females) 1990-92, ELT12 (Males), ELT14 (Males), ELT14 (Females), PA(90) (Males) and PA90(Females). In each case, the mode was found to the nearest half year of age simply by examining the life table d_{x }column, and the corresponding ageing parameter was either read directly from the _{x }column or obtained by log-linear interpolation from the _{x }column. The mode and ageing parameter of each of the above life tables are exhibited across the first two rows of table 1.

Using these parameters alone, estimates were prepared for the median, the lower quartile, the upper quartile, and the complete expectation of life at decennial ages under the assumption that mortality followed the Gompertz pattern. The formulae for the median, quartiles and expectation of life at birth were the exact formulae for these measures in respect of the Gompertz distribution over (-, ) and approximate therefore for the assumed Gompertz life table^{1}. The formula used for the complete expectation of life calculations for ages 40 and beyond is an approximate one derived explicitly for the Gompertz life table.